Combinatorial proofs for identities related to generalizations of the mock theta functions $ω(q)$ and $ν(q)$

Abstract

The two partition functions $p_ω(n)$ and $p_ν(n)$ were introduced by Andrews, Dixit and Yee, which are related to the third order mock theta functions $ω(q)$ and $ν(q)$, respectively. Recently, Andrews and Yee analytically studied two identities that connect the refinements of $p_ω(n)$ and $p_ν(n)$ with the generalized bivariate mock theta functions $ω(z;q)$ and $ν(z;q)$, respectively. However, they stated these identities cried out for bijective proofs. In this paper, we first define the generalized trivariate mock theta functions $ω(y,z;q)$ and $ν(y,z;q)$. Then by utilizing odd Ferrers graph, we obtain certain identities concerning to $ω(y,z;q)$ and $ν(y,z;q)$, which extend some early results of Andrews that are related to $ω(z;q)$ and $ν(z;q)$. In virtue of the combinatorial interpretations that arise from the identities involving $ω(y,z;q)$ and $ν(y,z;q)$, we finally present bijective proofs for the two identities of Andrews-Yee.